Signal compression is a common operation which is performed in many systems, including radar. The compression is often performed as a transform of domain, such as from the time domain to the frequency domain. The accuracy of the compression is limited by the finite amount of signal that can be collected. In the case of imaging radars, a signal consists of one or more sine waves in time that must be transformed into the spatial domain in order to determine their frequency, magnitude, and sometimes phase. The most common method for transformation is the Fourier transform.
The Fourier transform of a limited duration sine wave produces a waveform that can be described by a sinc function (FIG. 1). The sinc function has a mainlobe which contains the peak and has a width up to the first zero crossing, and a set of sidelobes comprising the oscillating remainder on both sides of the mainlobe. In radar and some other fields, the composite function of the mainlobe and the sidelobes is termed the impulse response (IPR) of the system. The location of the center of the sinc function is related to the frequency of the sine wave. If there are more than one sine wave present in the signal being analyzed, they will appear in the output at other locations. The resolution is related to the width of the mainlobe. The presence of sidelobes reduces the ability to discriminate between sinc functions.
Traditionally, the sidelobes of the impulse response have been reduced by multiplying the signal prior to compression by an amplitude function that is a maximum at the center and tending toward zero at the edges, as typified by a Hanning weighting function shown in FIG. 2. Sidelobe reduction by amplitude multiplication is called "weighting" or, sometimes, "apodization". Unfortunately, employing that kind of apodization to reduce sidelobes also results in the broadening of the mainlobe which degrades the resolution of the system.
One family of apodization functions is termed "cosine-on-pedestal". Hanning (50% cosine and 50% pedestal, as shown in FIG. 2) and Hamming (54% cosine and 46% pedestal) are two of the most popular. Hanning weighting reduces the peak sidelobe from -13 dB of the mainlobe's peak to -32 dB but it also doubles the mainlobe width (FIG. 3).
The equivalent of apodization can also be performed in the output domain by convolution. In the case of digitally sampled data, convolution is performed by executing the following operation on each point in the sequence: multiply each sample by a real-valued weight. which is dependent on the distance from the point being processed.
Any of the cosine-on-pedestal family of apodizations is especially easy to implement by convolution when the transform is of the same length as the data set, i.e., the data set is not padded with zeros before transformation. In this case, the convolution weights are non-zero only for the sample itself and its two adjacent neighbors. The values of the weights vary from [0.5, 1.0, 0.5] in the case of Hanning to [0.0, 1.0, 0.0] in the case of no apodization. Different cosine-on-pedestal apodization functions have different zero crossing locations for the sidelobes. The Hanning function puts the first zero crossing at the location of the second zero crossing of the unweighted impulse response. Not shown in FIGS. 1 and 3, the signs of the IPRs are opposite for all sidelobes when comparing unapodized and Hanning apodized signals.
To improve the process, a method called dual-apodization has been developed. In this method, the output signal is computed twice, once using no apodization and a second time using some other apodization which produces low sidelobes. Everywhere in the output, the two values are compared. The final output is always the lesser of the two. In this way the optimum mainlobe width is maintained while the sidelobes are generally lowered.
An extension to dual apodization is multi-apodization. In this method, a number of apodized outposts are prepared using a series of different apodizations, each of which have zero-crossings at different locations. The final output is the least among the ensemble of output apodized values at each output point. In the limit of an infinite number of apodizations, all sidelobes will be eliminated while the ideal mainlobe is preserved.
The final embodiment of this invention occurred when a method was discovered that could compute, for each sample in a sidelobe region, which of the cosine-on-pedestal functions provided the zero crossing from among the potentially infinite number of possible apodizations. This method is called spatially variant apodization (SVA). The method computes the optimum convolution weight set for each sample using a simple formula based on the value of the sample and two of its neighbors. Under noise free conditions, well separated compressed signals show only the mainlobes, and all sidelobes are removed. Under the usual noisy conditions, the output signal to background ratios are improved and the sidelobes are greatly reduced.